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Note that Byobu 5.12 uses these bindings by default for shift-left and shift-right.

And you can easily move a window from one window number to another by pressing:

ctrl-a :number [SOME_NUMBER]

However, GNU Screen does not support SOME_NUMBER to be a relative value, like +1 or -1. Rather, it must be an absolute window number, like "8" or "2".

Byobu also supports tmux as a backend, in addition to screen. Tmux does have support for this feature, and it can be accomplished by highlighting the window you want to move, and pressing Ctrl-Shift-F3 to move it left, and Ctrl-Shift-F4 to move it right.